
%!TEX program = xelatex
%!TEX TS-program = xelatex
%!TEX encoding = UTF-8 Unicode

\documentclass[10pt]{article} 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 一些常用的包总结在另一个文件里
\input{wang_preamble.tex}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 选择Windows操作系统写中文文档，使用 xelatex 或 lualatex 编译器
%\usepackage{xeCJK} % 处理中文、日文和韩文（统称为 CJK 文字）的排版
%\setCJKmainfont{SimSun} % 设置正文字体为宋体
%\setCJKmonofont{SimHei} % 设置加粗字体为黑体
%\setCJKsansfont{SimHei} % 黑体

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%选择Mac操作系统写中文文档，使用 xelatex 或 lualatex 编译器
\usepackage{xeCJK} % 支持中文字体
\setCJKmainfont{Songti SC} % 设置主要中文字体，用于正文中的中文文本。设置主要中文字体为宋体
%\setCJKmonofont{Menlo} % 设置等宽中文字体，用于代码块、等宽文本等。设置等宽中文字体为 Menlo
%\setCJKsansfont{PingFang SC} % 设置无衬线中文字体，用于标题、图表标签等。设置无衬线中文字体为 PingFang SC
%\setCJKromanfont{Songti SC} % 设置罗马中文字体，用于罗马字体中的中文文本。设置罗马中文字体为宋体

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%选择输出文档的两种类型：
%\newcommand{\showsolution}{0} %%设置showsolution=0, 编译生成试卷
\newcommand{\showsolution}{1} %%设置showsolution=1, 编译生成试卷解答

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 填写课程信息：
\newcommand{\CourseName}{复变函数}
\newcommand{\CourseStudents}{王立庆（2022 级数学与应用数学1班）}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%考完发给学生：

\usepackage{titling}
\setlength{\droptitle}{-2cm}   % 标题上移2cm


\ifnum\showsolution=0
\title{\CourseName \,\, 作业 }
\author{学号 \underline{\hspace{4cm}} \hspace{1cm} 姓名 \underline{\hspace{4cm}} }
\fi

\ifnum\showsolution=1
\title{\CourseName \,\, 解答 }
\author{\CourseStudents}
\fi

\renewcommand{\today}{\number\year \,年 \number\month \,月 \number\day \,日}
%\date{2023年4月24日}


%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%

\begin{document}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 给文档加标题、作者、日期、摘要

\maketitle

\thispagestyle{fancy} % 第一页也显示“第几页，共几页”的信息。

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\begin{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%\newpage 
\item %第3-3-3-E-7题
Find a linear transformation which carries $|z|=1$ and $|z-\frac{1}{4}|=\frac{1}{4}$ into concentric circles. What is the ratio of the radii? 

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%% 试卷留空位
\ifnum\showsolution=0
\vspace{3cm}
\fi

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%第3-3-3-E-7题解答
\ifnum\showsolution=1

{\color{red}Solution. 
\begin{enumerate}[label={\arabic*.}]
\item  
We start from two concentric circles $|w|=1$ and $|w|=R$, where $R>1$ is to be determined. 

\item  
We first use a translation $v=w+1$, thus the circle $|v-1|=1$ goes through the origin. 

\item  
We then use an inversion $\xi=\frac{1}{v}$, thus the circle that goes through the origin becomes a straight line $\mathrm{Re}(\xi)=\frac{1}{2}$. 
The other circle $|v-1|=R$ becomes another circle $|\xi-\frac{1}{1-R^2}|=\frac{-R}{1-R^2}$. 

\item  
We use a second translation $\eta=\xi -\frac{1}{1-R^2}$, to move the other circle's centre to the origin. 
Its equation is $|\eta|=\frac{-R}{1-R^2}$.  
The straight line $\mathrm{Re}(\xi)=\frac{1}{2}$ becomes the straight line $\mathrm{Re}(\eta)=\frac{1}{2}-\frac{1}{1-R^2}$. 

\item 
We use a homothetic transformation $u=\eta/k$, where $k$ is a constant to be determined. 
Our goal is to get the circle $|u|=1$ and the straight line $\mathrm{Re}(u)=2$. 
In oder to do this, we get a systems of two equations:
\begin{equation}
\left\{
\begin{aligned}
\frac{-R}{1-R^2} &= k, \\
\frac{1}{2} - \frac{1}{1-R^2} &= 2k.
\end{aligned}
\right. 
\end{equation}

\item 
Thus we solve $R=2+ \sqrt{3}$. 

\item 
We use another inversion $z=\frac{1}{u}$, the unit circle $|u|=1$ remains the unit circle $|z|=1$, and the straight line $\mathrm{Re}(u)=2$ becomes the circle $|z-\frac{1}{4}|=\frac{1}{4}$. 

\item 
Put all these transformations together, we get 
\begin{equation}
w=\frac{R-R^2z}{z-R}.
\end{equation}

\item  The graph of the these circles:

\begin{figure}[ht!]
    \centering
    \includegraphics[width=9cm, height=4.5cm]{exercise-3-3-3-E-7.png}
    \caption{A linear transformation carries two circles into concentric circles}
    \label{fig-3-3-3-7}
\end{figure}

\item  The python codes: 

\lstinputlisting{exercise-3-3-3-E-7.py}


\end{enumerate}
}



\vspace{0.2cm}

\fi

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
\end{enumerate}

%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%%
%%试卷结束
\end{document}
